def. Variance of a random variable X:

thm. (Variance Identities.)

  • For a single random variable and constant
    • Quadratic Scaling
    • Translation Invariance
  • For any two random variables :
  • …if (i.e. two are independent:
  • ….if Independent and Identically distributed (i.i.d) random variables

See also Covariance & Correlation.

def. Standard Deviation of a random variable is:

  • From the properties of variance we have


Motivation. Sometimes it’s nice to have random variables to have and . Using translation invariance and quadratic scaling we can take any random variable and standardize it.

thm. (Standardization) For random variable with and ,

  • Its expected value (mean) is 0; i.e.
  • Its standard deviation is 1; i.e.